SCIENTIA SINICA Informationis, Volume 50 , Issue 10 : 1501(2020) https://doi.org/10.1360/N112018-00260

A warning propagation algorithm to solve the double-objective minimum spanning tree

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  • ReceivedJan 16, 2019
  • AcceptedAug 13, 2019
  • PublishedOct 12, 2020


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  • Figure 1

    An instance of factor graph

  • Figure 2

    A model transformation instance of RBM

  • Figure 3

    Schematic diagram of transformation principle

  • Table 1   Parameters of 6 groups of random undirected graphs with different numbers of nodes
    $n$ $m$ $\alpha$ $d$
    100 4950 49.5 3.475
    150 11175 74.5 3.475
    200 19900 99.5 3.475
    300 44850 149.5 3.475
    350 61075 174.5 3.475
    400 79800 199.5 3.475

    Algorithm 1 Factor graph transformation algorithm

    Require:Random bigraph instance $G$;

    Output:Factor graph $F$;

    A random bigraph instance is selected;

    Select the initial point randomly $v\in~V$ as the starting point of transformation;

    while $v~\le~V$ do

    if $v$ is not the isolated point then

    The bigraph edge $e\in~E$ is mapped to a clause node of the factor graph;

    Two sets of weights on the edge of a bigraph are mapped to two nodes of a factor graph;


    Draw auxiliary edges to connect outliers;


    end if

    end while

    Add a binary tag bit to each edge and map it to the third argument node;


    Algorithm 2 Warning propagation algorithm for solving double objective minimum spanning tree

    Require:Factor graph $F$; Maximum number of iterations $t_{\rm~max}$; a requested precision $\varepsilon$;

    Output:The number of nodes in the minimum spanning tree $\eta~_{a\rightarrow~i}^{*}$;

    Get the factor graph by Algorithm 1;

    A node is randomly selected as the initial iteration node $n_{i}\in~V$;

    At time $t\leftarrow~0$, for every edge of the factor graph init the warning messages $W_{a\rightarrow~i}\in\left~\{~0,~1~\right~\}$;

    Let us do a random permutation of the edges in $G$;

    Using 2 to update the warning information until the algorithm converges;

    while $t~\le~t_{\rm~max}$ do

    Sweep the set of edges in the factor graph and update the warning information;

    if $\left~|\eta~_{a\rightarrow~i}(t)-\eta~_{a\rightarrow~i}(t-1)~~\right~|<~\varepsilon~$ then

    Takes the convergent nodes to form the nodes of the double-objective minimum spanning tree;



    end if


    end while

    The sum of the weights of two variable nodes representing weights in the factor graph is calculated respectively;

    Get part of the assignment;

  • Table 2   Comparison of the running time of MACS algorithm and WP algorithm (s)
    The number of node Enumeration MACS WP
    20 3360 552. 6 20. 423
    30 8700 681 30. 697
    40 20580 847. 2 40. 537
    50 45900 1095. 6 51. 018